I stumbled upon this https://en.wikipedia.org/wiki/Hopf_fibration through a Twitter clickhole and I’d like to know what I’m looking at.
Is it that a circle becomes a sphere by rotating (is that the right word?) in every direction? and then a sphere becomes a hypersphere by… being connected to every other part of its surface? What about a 4-sphere and a 5-sphere etc?
In: 1
A Hopf fibration is a way of cutting up a sphere in 4-dimension into circles, such that if you shrink all the circles each into a point you get a sphere in 3-dimension.
(terminology note: a sphere in 3-dimensional space is called a 2-sphere, a 2-dimensional sphere, because the actual surface of the sphere is only 2 dimensions; similarly sphere in 4-dimensional space)
Hopf fibration is extremely limited, only a few dimensions allow it.
An easy way to describe Hopf fibration is using complex numbers. Imagine a pair of complex number (z,w) whose sum of absolute value squared is 1: |z|^2 +|w|^2 =1. This is in the space of 4-dimension because each complex number is 2 dimensions. It’s a sphere because the radius is 1.
Now, cut this into circles as follow. We declare that any 2 pair of numbers (z1,w1) and (z2,w2) belongs to the same circle if there is a complex number t that when you multiply t by both number in a pair you get the other: z1=tz2, w1=tw2. It might be hard to see that they actually form circles, but after doing some algebra, you can check that t must always be a complex number of norm 1, so the possible value of t is indeed a circle. An alternative way of phrasing this is that the proportion z1:w1 and z2:w2 is the same; another way of phrasing this is that z1w2=z2w2 (by cross multiplication).
What if you shrink every circle into a point? You can see what shape it forms by compute z/w. Since every pair of complex numbers that form a circle have the same proportion, when you divide like this, you get a single point. Note that 1/0 is also a point, infinity. When you divide, you obtain every possible complex numbers together with an infinity all around you, and if you think about it (or use stereographic projection), this looks like a sphere.
This is a complicated abstract thing, so it will be more like ELI18. Let’s look at an easier example first.
If you can picture it, the entire plane can be decomposed as a bunch of vertical lines placed side-by-side. So to make the plane, what you can do is start with a fixed horizontal line (eg, the x-axis) and the place infinitely many vertical lines in a row on top of it. And you can take each of the vertical lines (fibers) and imagine squishing them down to just the point on the horizontal line. This is a “fibration”. You’re basically building the plane from “fibers” (vertical lines) placed at each point on our single horizontal line.
But all this is very simple and mathematicians like to ask the question: What if it was more complicated?
For instance, what if instead of a horizontal *line* we had a horizontal *circle*. That is, what if we place a vertical line “fiber” on each point of a circle? The resulting geometry would be a vertical cylinder. What if, further, we changed the geometry of the fibers themselves. Instead of putting *lines* at each point of the circle, what if we put vertical *circles* at each point on our horizontal circle? We’d get a torus (doughnut). You can imagine there being a circle on the inner-most spot on the doughnut hole, and circles radiating out of it, making the entire circle.
You can imagine, we can make some pretty wild shapes using this stuff. But, actually, these are still what we call “trivial”. You can think of “trivial” as being the most obvious way to paste vertical fibers onto our beginning “horizontal” shape. But there are other ways. As an example, let’s say we began to make the cylinder again but instead of just placing the vertical lines upright, what if we gave them a slight rotation. So, as you place a line, you give it a bit of a twist compared to the previous ones. Well, when you get back around, you will have twisted the whole thing and will have actually made a *[Mobius strip](https://en.wikipedia.org/wiki/Fiber_bundle#M%C3%B6bius_strip)*. Both the cylinder and Mobius strip are made from a “horizontal circle” and “vertical line fibers”, but the Mobius strip has some extra “stuff” that makes it more complicated.
So, in general, you can think of “Fibrations” as having three ingredients: 1.) A “horizontal” starting shape, 2.) a “vertical” fiber shape and 3.) how you decide “twist” the fibers together. Lots of different shapes can be made using these simple ingredients. You might try to think of some not mentioned here!
Now for Hopf Fibrations. If you start with a horizontal circle with vertical circle fibers, then the “trivial” fibration that you make is a torus and if you give it a “twist” then you actually make the [Klein Bottle](https://en.wikipedia.org/wiki/Fiber_bundle#Klein_bottle. Now, if you start with, instead, a horizontal *sphere* with vertical circle fibers then we can ask: What shapes can we make? The “trivial” shape will be a doughnut that lives in 4-dimensions that has a *spherical* hole rather than a circular one. A bit mindboggling, but still pedestrian. But, there is something special about these dimensions so that if you give it the right kind of twist, then you can actually use the sphere with circle fibers to make a hypersphere (a sphere in 4-dimensions). This is a bit surprising, because a hypersphere is a “nice” object, unlike a Klein Bottle or Mobius Strip which have funny properties.
What you see in [the picture on wikipedia](https://en.wikipedia.org/wiki/Hopf_fibration#/media/File:Hopf_Fibration.png) is actually a little harder to describe. The idea is that if you take a *few* points on the regular sphere and paste circles onto just those points, then you can get a slightly comprehensible image. The lower sphere shows the points on the sphere where it is attaching circles and the large image is the result. The moral is that these “folds” that you get are actually interwoven in a very complicated way so as to actually create the sphere without knotting it up as you might in a Klein bottle.
This kind of thing is actually *very* rare for this to happen, for you to be able to combine spheres of different dimensions like this to make more spheres. In fact, there are only *four* cases that you can do this and, surprisingly, these correspond to the kinds of “higher dimensional numbers” you can make. The simplest case of this gives you the ordinary real line. This case that we’re discussing is associated with the complex numbers. The next higher case gives you the Quaternions. The final case (which uses a “horizontal” hyperspheres in 9D with “vertical” fibers that are hyperspheres in 8D to construct a hypersphere in 16D) gives the [Octonions](https://en.wikipedia.org/wiki/Octonion).
Latest Answers